3.1 Basic modal analysis of acoustic barrier structures’ Vibration
According to the simulation method given in Section 2.1, the finite element model of the sound barrier is obtained, and the corresponding assembly drawings and grid drawings are shown in Fig. 1 (A) and (B), respectively. In order to explore the rationality of the model construction, the free vibration mode analysis of the sound barrier structure was first completed. The first five natural frequencies of the structure are 9.17, 16.1, 24.8, 31.9 and 45.8 Hz, respectively, and the first two mode shapes are shown in Fig. 1(C) and (D), and the first five mode shapes are shown in Figure S1 in the supporting file. It can be found that the first-order natural frequency of the sound barrier structure model established here is less than 10 Hz, which meets the requirements of the sound barrier design code, indicating that the model simplification assumption is reasonable and in line with reality. The calculated natural frequency can also be used as a reference for the comparison of the vibration mode changes of the sound barrier structure in other states.
3.2 Model analysis in different constrained state
By changing the constraints, the finite element model in the case of loose bolts of the sound barrier was simulated and analyzed. Firstly, the vibration mode analysis scenario of the sound barrier structure under the constraints of normal bolt tightening is established, as shown in Fig. 2(A). According to the actual situation of the construction site, the steel plate at the bottom of the sound barrier is fixed on the concrete foundation by four bolts, so the constraint at the bottom steel plate is arranged into a quartile part according to the bolt position distribution, and the five H-shaped steel columns involved in the model here are called C1 ~ C5 from right to left.
Table 2
The vibration frequency of the first 10 modes.
| Modal | Frequency(Hz) | Modal | Frequency(Hz) |
| Free Vibration | Normal Restraint Vibration | C2-3/4 Bolts Loosening | Free Vibration | Normal Restraint Vibration | C2-3/4 Bolts Loosening |
| 1st | 9.17 | 11.9 | 11.4 | 6th | 47.6 | 56.2 | 55 |
| 2nd | 16.1 | 22.2 | 21.5 | 7th | 63.1 | 60.9 | 60.8 |
| 3rd | 24.8 | 36 | 35.8 | 8th | 63.5 | 71.8 | 71.7 |
| 4th | 31.9 | 49.1 | 47.8 | 9th | 69.9 | 72.5 | 72.3 |
| 5th | 45.8 | 50.4 | 50.2 | 10th | 73.8 | 89.9 | 89.7 |
The simulation results of the vibration frequencies of the first 10 orders of the model are shown in Fig. 2(C) and Table 2, and the corresponding specific mode shapes can be found in S1 of the Supplementary Materials. Compared with the calculated natural frequency in the free state, the vibration frequency in the constrained state is almost all shifted to the right, which is in line with the objective trend of increasing the vibration frequency of the structure when the stiffness increases [15, 16].
The loosening of the steel plate bolts at the bottom of the sound barrier is simulated by removing the 3/4 constraints, and the simulation model (Fig. 2(B)) and mode shape calculation results (Fig. 2(D)) are given as an example of the loosening of 3 bolts out of 4 bolts. At the same time, the vibration frequencies and mode shapes of the first 10 order structures in this case are also compared in S2 in the Supplementary Materials, respectively. It can be seen that when the 3/4 bolts of the C2 column are loosened and no longer tightened, the mode shapes of the whole structure do not change much, but the vibration form changes from the original symmetrical distribution to asymmetry. In terms of frequency, the vibration frequency of each order is slightly reduced, the reduction amplitude is not more than 5%, and the change is basically not more than 1 Hz. From the perspective of the displacement change amplitude, the loosening of the bolt will lead to an increase in the displacement change of the acoustic insulation panel, with a maximum increase of about 0.32 mm (fifth-order mode shape), as shown in Fig. 2(D). The result shows that the incomplete loosening state of the local bolt has little influence on the mode of the overall structure, and the effect of judging the bolt state of one of the column bases by observing the mode shape change may be limited.
3.3 Modal analysis of bolts on different columns with loosening conditions
According to the results of Section 3.2, when some bolts at the bottom of the S2 column are loosened, the structural mode shape, frequency and displacement amplitude of the overall sound barrier do not change significantly. If the sensor is used for monitoring in the actual project, the amount of variation in the monitoring result may be less than the accuracy or measurement error of the sensor itself. In order to study the influence of the bottom constraint on the vibration response of the sound barrier, the bottom constraint was completely removed in the subsequent simulation. At the same time, in order to study whether the influence of different columns is the same, the structural modal analysis of different column bolts under the condition of complete loosening of bolts is carried out here. The detail simulation methods are shown in S3, and the results of the first 10 order modes are detailed in S2 in the supporting materials.
Part of the finite element simulation results are shown in Fig. 3, and the rest of the mode shape simulation results are given in full in S2 of the support material. By comparing the mode shapes of the normal restraint of the outermost column (C1) (Fig. 3(A)) and the bottom bolt after the bottom bolt is completely loosened (Fig. 3(B)), it can be seen that after C1 completely loses the bottom restraint, the first five vibration frequencies of the overall sound barrier change most obviously, and a new structural mode shape (at 10 Hz) appears.
Comparing the mode shapes and frequencies of the normal state, C1, C2 and C3 columns when the bottom of the column is loose, it is found that when the C2 column on the left and the C3 column in the center are loose, the vibration frequency of the second order to the fourth order changes significantly. In the third-order mode shapes shown in Fig. 3(C)-(F), the frequency changes from 36 Hz (normal state) to 24 Hz (C1 bottom unconstrained), 32.3 Hz (C2 bottom unconstrained), and 29.9 Hz (C3 bottom unconstrained), respectively. This shows that the overall mode of the structure is sensitive to the complete loosening of the bottom of a single column, and taking the third-order mode shape as an example, the new frequency of the whole structure will appear, and the loose part will also produce a local mode. At the same time, the 6 ~ 10th order mode shape also indicates that the structure produces a new local mode in the loose position, which can be seen in the S2 in the Supplementary Materials. It is speculated that this is due to a subtle change in dynamics between the fasteners when the bolts used to fasten the connection between the column and the concrete foundation come loose [17].
From the above mode shape simulation results, it can be seen that when the bolt is loosened, a new local mode will appear nearby. This conclusion suggests that if you want to monitor changes in vibration response with sensors, you can install them near the bolts and tell if the bolts are loose by using new local response changes. The top of the column is usually the most significant position of vibration, whether it is the loosening of its own bolts or the loosening of the bolts of adjacent columns, there are great changes, and it can also be used as a good sensor to monitor the abnormal vibration of the sound barrier caused by the loosening of bolts.
3.4 Simulation of vibration response at different positions of sound barrier columns
During the service of the high-speed railway sound barrier, the sound barrier will be subjected to the load perpendicular to the column caused by the train wind, resulting in vibration of the overall structure [18, 19]. Once the vibration displacement of the sound barrier exceeds the threshold, it will cause an accident and affect the safe operation of the train. As a result, the possible vibrational displacement of the sound barrier has also become the focus of attention. Here, in the simulation model, the bolt loosening scenes at different positions (the selected positions are C1, C2, and the bottom of the C3 column) are constructed, and the force of 1000N is selected as the load condition, which is loaded in the center of the C3 column (refer to S4 in the support material for details). Then, the simulation analysis sites were set up at the left and right symmetry of the forced columns, the adjacent columns, and the interval columns, and the overall vibration deformation response of the sound barrier was simulated and calculated, and the simulation results obtained are shown in Fig. 4.
The results show that when the force of 1000N is used as the percussion force to hit the central column, when the bottom of the central column (C3) is loose, the displacement of the top of the column with the largest displacement change will increase by 30% compared with the normal constraint. If the bottom of the column (C2) is loosened, the displacement of the monitoring point increases by 10%, and if the bottom of the column (C1) is faulty, the displacement of the monitoring point hardly increases. The results show that when the column vibrates, the vibration will be transmitted to other adjacent columns, but only the displacement of the top of the nearest column can change significantly, and with the increase of distance, the vibration will attenuate [20] and no longer cause the vibration displacement of the column farther away. This also serves as an indication in engineering applications that sensors can be placed at maximum interval of one column when sensors are placed to monitor abnormal vibrations caused by the bottom loosen bolts.